Which equation can be used to solve for c? Triangle A B C is shown. Angle A C B is 90 degrees and angle A B C is 35 degrees. The length of C B is 5 inches, the length of A C is b, and the length of B A is c.
The Correct Answer and Explanation is:
To solve for ccc in triangle ABC, where:
- ∠ACB=90∘\angle ACB = 90^\circ∠ACB=90∘ (a right triangle),
- ∠ABC=35∘\angle ABC = 35^\circ∠ABC=35∘,
- CB=5CB = 5CB=5 inches (the length of the side opposite angle A),
- AC=bAC = bAC=b inches (the length of the side opposite angle B),
- AB=cAB = cAB=c inches (the hypotenuse).
We can apply trigonometric functions like sine, cosine, and Pythagoras’ theorem to solve for ccc.
Step 1: Use the cosine function.
Since ∠ABC=35∘\angle ABC = 35^\circ∠ABC=35∘ and ABABAB is the hypotenuse of the right triangle, we can use the cosine function to relate ccc to the sides:cos(∠ABC)=adjacenthypotenuse=ACAB=bc\cos(\angle ABC) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AC}{AB} = \frac{b}{c}cos(∠ABC)=hypotenuseadjacent=ABAC=cb
This gives us the equation:cos(35∘)=bc\cos(35^\circ) = \frac{b}{c}cos(35∘)=cb
Rearranging to solve for ccc, we get:c=bcos(35∘)c = \frac{b}{\cos(35^\circ)}c=cos(35∘)b
Step 2: Use the Pythagorean theorem for verification.
Since triangle ABC is a right triangle, you can also use the Pythagorean theorem to find the relationship between the sides:AB2=AC2+BC2AB^2 = AC^2 + BC^2AB2=AC2+BC2
Substituting the known values, we get:c2=b2+52c^2 = b^2 + 5^2c2=b2+52
So, the second equation you can use is:c=b2+25c = \sqrt{b^2 + 25}c=b2+25
Conclusion:
You can solve for ccc using either the equation c=bcos(35∘)c = \frac{b}{\cos(35^\circ)}c=cos(35∘)b (from trigonometry) or c=b2+25c = \sqrt{b^2 + 25}c=b2+25 (from the Pythagorean theorem). Each approach gives you a way to express ccc in terms of bbb, depending on the available information in the problem.
